Finite Group Databasebeta

← #238771

Subgroups — #238771

Order 512.

980 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37641C211yesmaximal, characteristic
2#21860C212yesmaximal
3#20800C211yesmaximal, characteristic
4C2×#1485C212yesmaximal
5#10439C214yesmaximal, complemented
6#10437C214yesmaximal, complemented
7#8038C211yesmaximal, characteristic
8C2×#533C2212yes—
9C2×#431C2212yes—
10C2×#428—22no—
11C2×C4×M16C2211yescharacteristic
12#2822—22no—
13#2625—24no—
14#2621—22no—
15#2582C2212yes—
16#2582C2212yes—
17#2568—22no—
18#2546—22no—
19#2285C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#1990—22no—
22#1956—22no—
23#1872—24no—
24#1847—22no—
25#1845C2212yes—
26#1844C2212yes—
27#1803C2218yes—
28#1485—28no—
29C2×Q8⋊C8C2218yes—
30#1296C2212yes—
31#1296C2212yes—
32C22×M16—21no—
33#533—24no—
34#533—24no—
35#533—24no—
36#533—24no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8D412yes—
40C2×#125—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8D414yes—
43C2×C4×Q8—22no—
44C2×C4×Q8C2314yes—
45C2×#117—22no—
46C2×#117D412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C42C2311yescharacteristic
50#431—24no—
51#431—24no—
52#428—48no—
53C2×#107—24no—
54C2×#106—24no—
55C2×C4⋊C8C2314yes—
56C2×#103—28no—
57C2×C22⋊C8—22no—
58C4×M16—22no—
59C4×M16D414yes—
60C2×#97—21no—
61C2×C4×C8—22no—
62C2×C4×C8C2312yes—
63#400—28no—
64#398—28no—
65#398—24no—
66#398—24no—
67#397—24no—
68#397—24no—
69#391—24no—
70#391—24no—
71#391—24no—
72#389—24no—
73#389—24no—
74#384—21no—
75#384—22no—
76#384C2312yes—
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#382—22no—
80#382D412yes—
81#380—22no—
82C4×C4⋊C4—22no—
83#340—28no—
84Q8⋊C8—216no—
85C22×Q8—22no—
86C22×Q8C2×D414yes—
87C23×C4—41no—
88C23×C4C2×D412yes—
89C23×C4C2×D411yescharacteristic
90C2×M16—24no—
91C2×M16—24no—
92C2×M16—24no—
93C22×C8—22no—
94C22×C8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—24no—
99C4⋊Q8—84no—
100#125—44no—
101#125—44no—
102C22⋊Q8—28no—
103C22⋊Q8—28no—
104C22⋊Q8—44no—
105C22⋊Q8—44no—
106C4×Q8—24no—
107C4×Q8—24no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—44no—
111C4×Q8—44no—
112#117—44no—
113#117—44no—
114#117—44no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—84no—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—22no—
135C2×C42—22no—
136C2×C42—24no—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C2×C42C2×D411yescharacteristic
140C2×C42#4112yes—
141C2×C42#4114yes—
142C2×C42C2411yescharacteristic
143#107—48no—
144#106—48no—
145C4⋊C8—24no—
146C4⋊C8—24no—
147#103—88no—
148C22⋊C8—44no—
149#97—24no—
150C4×C8—22no—
151C4×C8—44no—
152C4×C8C2×D414yes—
153#95—24no—
154#95—24no—
155#95—28no—
156#95—28no—
157#95—24no—
158C24#12011yescharacteristic
159C2×Q8—24no—
160C2×Q8—24no—
161C2×Q8—28no—
162C2×Q8—28no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—44no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—42no—
175C22×C4—82no—
176C22×C4—84no—
177C22×C4#12012yes—
178C22×C4#12011yescharacteristic
179C22×C4#12114yes—
180C22×C4#12112yes—
181C22×C4#12314yes—
182C22×C4#12412yes—
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D412yes—
185M16—216no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8—44no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—48no—
204C4⋊C4—44no—
205C4⋊C4—48no—
206C4⋊C4—84no—
207C4⋊C4—84no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C22⋊C4—48no—
211C22⋊C4—82no—
212C42—44no—
213C42—44no—
214C42—44no—
215C42#12114yes—
216C42#12112yes—
217C42#12314yes—
218C42#12412yes—
219C42C22×D411yescharacteristic
220C42C2×#4111yescharacteristic
221C42C2×#4112yes—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23#39412yes—
227C23C2×#12011yescharacteristic
228Q8—48no—
229Q8—48no—
230Q8—88no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—28no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—28no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—164no—
252C2×C4#39414yes—
253C2×C4#39412yes—
254C2×C4#45312yes—
255C2×C4#45612yes—
256C2×C4C2×#12111yescharacteristic
257C2×C4C2×#12112yes—
258C2×C4C2×#12312yes—
259C2×C4C2×#12411yescharacteristic
260C2×C4#57714yes—
261C2×C4#57814yes—
262C8—28no—
263C8—84no—
264C22—24no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22C2×#39411yescharacteristic
270C22#301012yes—
271C22#301112yes—
272C22C2×#45311yescharacteristic
273C22C2×#45611yescharacteristic
274C4—24no—
275C4—21no—
276C4—21no—
277C4—82no—
278C4—82no—
279C4—84no—
280C4—84no—
281C4—162no—
282C4#203914yes—
283C4#306314yes—
284C4#309714yes—
285C2—42no—
286C2#1871811yescharacteristic
287C2#1871911yescharacteristic
288C2#1873712yes—
289C2#2348412yes—
290C2C2×#203911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #10439C2semidirect
6. #10437C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#376412
#218602
#208002
C2×#14852
#104392
#104372
#80382